Recently Guth and Katz \cite{GK2} invented, as a step in their nearly complete solution of Erd\H{o}s's distinct distances problem, a new method for partitioning finite point sets in $\R^d$, based on the Stone--Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemer\'edi--Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach--Sharir bound on the number of incidences with algebraic curves of bounded degree.

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth--Katz Polynomial Partitioning Technique

We show that the number of unit distances determined by n points in â3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].

/pdf/unit-distances-in-three-dimensions-3ctoxp79dk.pdf

Unit distances in three dimensions

Recently Guth and Katz ( arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in ℝ d , based on the Stone–Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemeredi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.

/pdf/simple-proofs-of-classical-theorems-in-discrete-geometry-via-4ph619hp1v.pdf

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Factorial k-means (FKM) clustering is a method for clustering objects in a low-dimensional subspace. The advantage of this method is that the partition of objects and the low-dimensional subspace reflecting the cluster structure are obtained, simultaneously. Conditions that ensure the almost sure convergence of the estimator of FKM clustering as the sample size increases unboundedly are derived. The result is proved for a more general model including FKM clustering.

Strong Consistency of Factorial K-means Clustering

The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set $X$ is a set of polygonal curves in $\mathbb{R}^d$ and the sets $\mathcal{R}$ are metric balls defined by curve similarity metrics, such as the Frechet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

The VC Dimension of Metric Balls under Fr\'echet and Hausdorff Distances

Motivated by statistical learning theoretic treatment of principal component analysis, we are concerned with the set of points in ℝd that are within a certain distance from a k-dimensional affine subspace. We prove that the VC dimension of the class of such sets is within a constant factor of (k+1)(d−k+1), and then discuss the distribution of eigenvalues of a data covariance matrix by using our bounds of the VC dimensions and Vapnik’s statistical learning theory. In the course of the upper bound proof, we provide a simple proof of Warren’s bound of the number of sign sequences of real polynomials.

/pdf/vc-dimensions-of-principal-component-analysis-599h8r801w.pdf

Yasutaka Uwano

Papers

VC Dimensions of Principal Component Analysis